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Related concepts
The circle group is equivalently (isomorphically)
the quotient group of the additive group of real numbers by the additive group of integers, induced by the canonical embedding ;
the unitary group ;
the special orthogonal group ;
the subgroup of the group of units of the field of complex numbers (its multiplicative group) given by those of any fixed positive modulus (standardly ).
For general abstract properties usually the first characterization is the most important one. Notably it implies that the circle group fits into a short exact sequence
the “real exponential exact sequence”.
(On the other hand, the last characterization is usually preferred when one wants to be concrete.)
A character of an abelian group is simply a homomorphism from to the circle group.
is the compact real form of the multiplicative group over the complex numbers, see at form of an algebraic group – Circle group and multiplicative group.
A principal bundle with structure group the circle group is a circle bundle. The associated bundle under the standard linear representation of is a complex line bundle.
rotation groups in low dimensions:
see also
Last revised on July 12, 2021 at 18:34:58. See the history of this page for a list of all contributions to it.